Graphs with girth $2\mathrm{\ell }+1$ and without longer odd holes are $3$-colorable

For a number $\mathrm{\ell }\ge 2$, let ${\mathcal{G}}_{\mathrm{\ell }}$ denote the family of graphs which have girth $2\mathrm{\ell }+1$ and have no odd hole with length greater than $2\mathrm{\ell }+1$. Plummer and Zha conjectured that every 3-connected and internally 4-connected graph in ${\mathcal{G}}_2$ is 3-colorable. Wu, Xu, and Xu conjectured that every graph in ${\bigcup }_{\mathrm{\ell }\ge 2}{\mathcal{G}}_{\mathrm{\ell }}$ is 3-colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in ${\mathcal{G}}_2$ and ${\mathcal{G}}_3$ is 3-colorable. In this paper, we prove that every graph in ${\bigcup }_{\mathrm{\ell }\ge 5}{\mathcal{G}}_{\mathrm{\ell }}$ is 3-colorable.